Edexcel F2 (Further Pure Mathematics 2) 2023 June

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. Show that, for \(r \geqslant 2\) $$\frac { 2 } { \sqrt { r } + \sqrt { r - 2 } } = \sqrt { r } - \sqrt { r - 2 }$$
2. Hence use the method of differences to determine $$\sum _ { r = 2 } ^ { n } \frac { 2 } { \sqrt { r } + \sqrt { r - 2 } }$$ giving your answer in simplest form.
3. Hence show that $$\sum _ { r = 4 } ^ { 50 } \frac { 2 } { \sqrt { r } + \sqrt { r - 2 } } = A + B \sqrt { 2 } + C \sqrt { 3 }$$ where \(A\), \(B\) and \(C\) are integers to be determined.

1. The complex number \(z _ { 1 }\) is defined as

$$z _ { 1 } = \frac { \left( \cos \frac { 5 \pi } { 12 } + i \sin \frac { 5 \pi } { 12 } \right) ^ { 4 } } { \left( \cos \frac { \pi } { 3 } - i \sin \frac { \pi } { 3 } \right) ^ { 3 } }$$

1. Without using your calculator show that $$z _ { 1 } = \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 }$$
2. Shade, on a single Argand diagram, the region \(R\) defined by $$\left| z - z _ { 1 } \right| \leqslant 1 \quad \text { and } \quad 0 \leqslant \arg \left( z - z _ { 1 } \right) \leqslant \frac { 3 \pi } { 4 }$$ Given that the complex number \(z\) lies in \(R\)
3. determine the smallest possible positive value of \(\arg z\)

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} Given that $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { k ( x - 1 ) }$$ where \(k\) is a positive constant,

1. show that $$( x + 4 ) ( x - 1 ) \left( p x ^ { 2 } + q x + r \right) \leqslant 0$$ where \(p , q\) and \(r\) are expressions in terms of \(k\) to be determined.
2. Hence, or otherwise, determine the values for \(x\) for which $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { 3 ( x - 1 ) }$$

1. (a) Determine the general solution of the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 48 x ^ { 2 } - 34$$ Given that \(y = 4\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 21\) at \(x = 0\)
(b) determine the particular solution of the differential equation.
(c) Hence find the value of \(y\) at \(x = - 2\), giving your answer in the form \(p \mathrm { e } ^ { q } + r\) where \(p , q\) and \(r\) are integers to be determined.

1. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\) is given by

$$w = \frac { z + 1 } { z - 3 } \quad z \neq 3$$ The straight line in the \(z\)-plane with equation \(y = 4 x\) is mapped by \(T\) onto the circle \(C\) in the \(w\)-plane.

1. Show that \(C\) has equation $$3 u ^ { 2 } + 3 v ^ { 2 } - 2 u + v + k = 0$$ where \(k\) is a constant to be determined.
2. Hence determine
   1. the coordinates of the centre of \(C\)
   2. the radius of \(C\)

1. Given that \(y = \sec x\)
   1. show that
   $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \sec x \tan x \left( p \sec ^ { 2 } x + q \right)$$ where \(p\) and \(q\) are integers to be determined.
2. Hence determine the Taylor series expansion about \(\frac { \pi } { 3 }\) of sec \(x\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\), up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\), giving each coefficient in simplest form.
3. Use the answer to part (b) to determine, to four significant figures, an approximate value of \(\sec \left( \frac { 7 \pi } { 24 } \right)\)

1. (a) Show that the substitution \(z = y ^ { - 2 }\) transforms the differential equation

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + y + 4 x ^ { 2 } y ^ { 3 } \ln x = 0 \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - \frac { 2 z } { x } = 8 x \ln x \quad x > 0$$ (b) By solving differential equation (II), determine the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\)

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$r = 6 ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$ Given that \(C\) meets the initial line at the point \(A\), as shown in Figure 1,

1. write down the polar coordinates of \(A\). The line \(l _ { 1 }\), also shown in Figure 1, is the tangent to \(C\) at the point \(B\) and is parallel to the initial line.
2. Use calculus to determine the polar coordinates of \(B\). The line \(l _ { 2 }\), also shown in Figure 1, is the tangent to \(C\) at \(A\) and is perpendicular to the initial line. The region \(R\), shown shaded in Figure 1, is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\)
3. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(p \sqrt { 3 } + q \pi\) where \(p\) and \(q\) are constants to be determined.